Mathematics of Ringiana and Ringsanity II: Solving x + x = x

This post continues part I: Moves of self-similarity.

Ringiana (for Android and iPhone), Ringsanity (for iPhone), and the math behind these games are based on the idea of manipulating objects that are identical to two copies of themselves. Imagine an abstract object X that can be identified with the double of itself. For lack of notation and clear notion of doubling an object we are going to be informal and denote the double by X+X. Then we are looking for X such that 

X + X = X 

Imagine that X, however abstract, admits some tangible invariant or measurement, such as size, area, length, or volume. Denote the invariant by x. Presumably, the equation X + X = X should translate into the corresponding property of the invariant 

x + x = x 

Say x is a number, then we are trying to solve the equation

2x = x 

Cracking it is not too hard, but, sadly, gives us a not-too-useful unique solution 

x = 0  

telling us that the measurement is zero. At this point it's ok to give up - our object X will have no useful measurements or nontrivial invariants to show. Or we can become creative and set x to infinity 

x = ∞   

Not trying to be too clever, let's think of infinity as a very large number, or better a sequence of numbers that get bigger and bigger. If we can operate on infinity as on a usual number, adding ∞ and a number should be possible - and the answer is easy to guess: 

  ∞ + 10 = ∞ 

and, more generally,

∞ + n = ∞

for any number n. Making n larger and larger, we should get

  ∞ + ∞  = ∞

and suddenly rejoyce in our Eureka moment -- a nontrivial solution is at hand, for our mysterious measurement x is infinity: 

x =  ∞

Whether infinity can be treated as a legitimate number is a tricky question. Examine the simplest popular number system - integers, usually denoted Z. You can add and subtract integers, with these operations satisfying lots of important properties (axioms). Once you add infinity to Z, the structural beauty of integers breaks down. What is ∞ - ∞, infinity minus infinity? It can potentially be anything. Both infinities in the expression 

∞ -  ∞

can be thought of as limits of large numbers. If I choose the sequence of increasing numbers 1,2,3,4, ... for the first infinity and a faster growing sequence 1,2,4,8,16,...,2048,... for the second, hopefully the difference ∞ - ∞ should correspond to the sequence of differences: 

1 - 1 = 0,  2 - 2 = 0, 3 - 4 = -1, 4 - 8 = -4, ..., 

10 - 2048 = - 2038, ... ,

a sequence of negative numbers which should describe -∞, minus infinity. Do we get that ∞ - ∞ = - ∞ ? Not really, for if we swap the sequences and examine the differences, the numbers will be positive and large, and one can as well say that ∞ - ∞ = ∞. With a tiny amount of extra work you can set up the sequences to get ∞ - ∞ = n, for any number n. The difference of two infinities behaves pretty democratically and can take on any value. This teaches us to treat infinities carefully and don't extend the familiar operations on numbers to them in a cavalier way.

Allowing infinity into our world to get a nontrivial solution to the equation 

 x + x = x 

is not a bad idea, however. Something has to give, and we choose to disallow subtraction from our system. Let's keep addition and multiplication, to at least have something. Without subtraction, we can as well downsize from integers to natural numbers, denoted N. They are 0, 1, 2, 3, 4, ... Due to historical and cultural differences in teaching mathematics, some people consider 0 a natural number and some do not. You do get a richer structure if you include 0, and we follow this convention. 

After adding infinity ∞ to natural numbers N both addition and multiplication can still be defined on this larger system and will share many properties with the corresponding operations on natural numbers. Some other properties will fail, though, and the union of infinity and natural numbers is not a particularly beautiful mathematical structure.   

Luckily, the day can be saved: the union of natural numbers and infinity is a shadow of a better and healthier structure that will be revealed in the next installment of the series.